Algebra

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Smarandache Idempotents in Loop Rings ZtLn(m) of the Loops Ln(m):

In this paper we establish the existance of S-idempotents in case of loop rings
ZtLn(m) for a special class of loops Ln(m); over the ring of modulo integers
Zt for a specific value of t. These loops satisfy the conditions gi2 = 1 for every
gi ε Ln(m). We prove ZtLn(m) has an S-idempotent when t is a perfect number
or when t is of the form 2ip or 3ip (where p is an odd prime) or in general when
t = p1ip2 (p1 and p2 are distinct odd primes). It is important to note that we
are able to prove only the existance of a single S-idempotent; however we leave
it as an open problem wheather such loop rings have more than one S-idempotent.
This paper has three sections. In section one, we give the basic notions about
the loops Ln(m) and recall the definition of S-idempotents in rings. In section
two, we establish the existance of S-idempotents in the loop ring ZtLn(m). In
the final section, we suggest some interesting problems based on our study.
Category:Algebra